In the manufacture of tube and fin heat exchanger cores having elongated reinforcements that extend along opposite sides of the core and are brazed together with the tubes to headers at opposite ends of the core, it has been the practice to flare the tube ends where they extend past the headers to ensure that the tube to header joints are subsequently brazed. In addition, one or more detachable metal bands are wrapped around the core assembly to hold the reinforcements in place for the brazing. With the core assembly thus held together by the flared tube ends and band, it is then normally brazed with the tubes in the vertical position by being supported at the upper header thereof. As a result, the core weight is supported only by the flared tube ends and it has been found that increases in core assembly size can create problems of tube slip where the tubes actually drop from the upper header during the braze cycle because of softening of the weight supporting flared tube ends and in addition, unbonded reinforcement to header joints because of the separation that then occurs between the reinforcements and the header plates.